diff --git a/src/plonk.rs b/src/plonk.rs
index 7844d287..bb00b2c6 100644
--- a/src/plonk.rs
+++ b/src/plonk.rs
@@ -5,11 +5,14 @@
 //! [halo]: https://eprint.iacr.org/2019/1021
 //! [plonk]: https://eprint.iacr.org/2019/953
 
-use crate::arithmetic::CurveAffine;
+use blake2b_simd::Params as Blake2bParams;
+
+use crate::arithmetic::{CurveAffine, FieldExt};
 use crate::poly::{
     commitment::Params, Coeff, EvaluationDomain, ExtendedLagrangeCoeff, LagrangeCoeff, Polynomial,
 };
 use crate::transcript::{ChallengeScalar, Transcript};
+use std::convert::TryInto;
 
 mod circuit;
 mod keygen;
@@ -77,13 +80,38 @@ impl<C: CurveAffine> VerifyingKey<C> {
 
     /// Hashes a verification key into a transcript.
     pub fn hash<T: Transcript<C>>(&self, transcript: &mut T) -> io::Result<()> {
+        let mut hasher = Blake2bParams::new()
+            .hash_length(64)
+            .personal(C::BLAKE2B_PERSONALIZATION)
+            .to_state();
+
+        // Hash in constants in the domain which influence the proof
+        let domain_hash = &self.domain.hash(&mut hasher);
+        transcript.common_scalar(C::Scalar::from_bytes_wide(domain_hash))?;
+
+        // Hash in `ConstraintSystem`
+        let cs_hash = &self.cs.hash(&mut hasher);
+        transcript.common_scalar(C::Scalar::from_bytes_wide(cs_hash))?;
+
+        // Hash in vector of fixed commitments
+        hasher.update(b"num_fixed_commitments");
+        hasher.update(&self.fixed_commitments.len().to_le_bytes());
         for commitment in &self.fixed_commitments {
             transcript.common_point(*commitment)?;
         }
+
+        // Hash in vector of permutation arguments
+        hasher.update(b"num_permutations");
+        hasher.update(&self.permutations.len().to_le_bytes());
         for permutation in &self.permutations {
             permutation.hash(transcript)?;
         }
 
+        // Hash in final Blake2bState
+        transcript.common_scalar(C::Scalar::from_bytes_wide(
+            hasher.finalize().as_bytes().try_into().unwrap(),
+        ))?;
+
         Ok(())
     }
 }
diff --git a/src/plonk/circuit.rs b/src/plonk/circuit.rs
index e8dfcfcc..020ee35d 100644
--- a/src/plonk/circuit.rs
+++ b/src/plonk/circuit.rs
@@ -1,8 +1,9 @@
+use blake2b_simd::State as Blake2bState;
 use core::cmp::max;
 use core::ops::{Add, Mul};
 use ff::Field;
 use std::{
-    convert::TryFrom,
+    convert::{TryFrom, TryInto},
     ops::{Neg, Sub},
 };
 
@@ -10,7 +11,7 @@ use super::{lookup, permutation, Error};
 use crate::poly::Rotation;
 
 /// A column type
-pub trait ColumnType: 'static + Sized {}
+pub trait ColumnType: 'static + Sized + std::fmt::Debug {}
 
 /// A column with an index and type
 #[derive(Clone, Copy, Debug, Eq, PartialEq, Hash)]
@@ -27,6 +28,11 @@ impl<C: ColumnType> Column<C> {
     pub(crate) fn column_type(&self) -> &C {
         &self.column_type
     }
+
+    pub(crate) fn hash(&self, hasher: &mut Blake2bState) -> [u8; 64] {
+        hasher.update(&format!("{:?}", self).as_bytes());
+        hasher.finalize().as_bytes().try_into().unwrap()
+    }
 }
 
 /// An advice column
@@ -317,6 +323,12 @@ impl<F: Field> Expression<F> {
             Expression::Scaled(poly, _) => poly.degree(),
         }
     }
+
+    /// Hash an Expression into a Blake2bState
+    pub fn hash(&self, hasher: &mut Blake2bState) -> [u8; 64] {
+        hasher.update(&format!("{:?}", self).as_bytes());
+        hasher.finalize().as_bytes().try_into().unwrap()
+    }
 }
 
 impl<F: Field> Neg for Expression<F> {
@@ -598,4 +610,70 @@ impl<F: Field> ConstraintSystem<F> {
         self.num_instance_columns += 1;
         tmp
     }
+
+    /// Hashes the `ConstraintSystem` into a `u64`.
+    pub fn hash(&self, mut hasher: &mut Blake2bState) -> [u8; 64] {
+        hasher.update(b"num_fixed_columns");
+        hasher.update(&self.num_fixed_columns.to_le_bytes());
+
+        hasher.update(b"num_advice_columns");
+        hasher.update(&self.num_advice_columns.to_le_bytes());
+
+        hasher.update(b"num_aux_columns");
+        hasher.update(&self.num_aux_columns.to_le_bytes());
+
+        hasher.update(b"num_gates");
+        hasher.update(&self.gates.len().to_le_bytes());
+        for gate in self.gates.iter() {
+            hasher.update(gate.0.to_owned().as_bytes());
+            gate.1.hash(&mut hasher);
+        }
+
+        hasher.update(b"num_advice_queries");
+        hasher.update(&self.advice_queries.len().to_le_bytes());
+        for query in self.advice_queries.iter() {
+            query.0.hash(&mut hasher);
+            query.1.hash(&mut hasher);
+        }
+
+        hasher.update(b"num_aux_queries");
+        hasher.update(&self.aux_queries.len().to_le_bytes());
+        for query in self.aux_queries.iter() {
+            query.0.hash(&mut hasher);
+            query.1.hash(&mut hasher);
+        }
+
+        hasher.update(b"num_fixed_queries");
+        hasher.update(&self.fixed_queries.len().to_le_bytes());
+        for query in self.fixed_queries.iter() {
+            query.0.hash(&mut hasher);
+            query.1.hash(&mut hasher);
+        }
+
+        hasher.update(b"num_permutations");
+        hasher.update(&self.permutations.len().to_le_bytes());
+        for argument in self.permutations.iter() {
+            hasher.update(&argument.get_columns().len().to_le_bytes());
+            for column in argument.get_columns().iter() {
+                column.hash(&mut hasher);
+            }
+        }
+
+        hasher.update(b"num_lookups");
+        hasher.update(&self.lookups.len().to_le_bytes());
+        for argument in self.lookups.iter() {
+            hasher.update(&argument.input_columns.len().to_le_bytes());
+            hasher.update(&argument.table_columns.len().to_le_bytes());
+            for (input, table) in argument
+                .input_columns
+                .iter()
+                .zip(argument.table_columns.iter())
+            {
+                input.hash(&mut hasher);
+                table.hash(&mut hasher);
+            }
+        }
+
+        hasher.finalize().as_bytes().try_into().unwrap()
+    }
 }
diff --git a/src/poly.rs b/src/poly.rs
index 5ac1462c..45088f4e 100644
--- a/src/poly.rs
+++ b/src/poly.rs
@@ -4,7 +4,9 @@
 
 use crate::arithmetic::parallelize;
 
+use blake2b_simd::State as Blake2bState;
 use ff::Field;
+use std::convert::TryInto;
 use std::fmt::Debug;
 use std::marker::PhantomData;
 use std::ops::{Add, Deref, DerefMut, Index, IndexMut, Mul, RangeFrom, RangeFull, Sub};
@@ -228,4 +230,10 @@ impl Rotation {
     pub fn next() -> Rotation {
         Rotation(1)
     }
+
+    /// Hash Rotation into a Blake2bState
+    pub fn hash(&self, hasher: &mut Blake2bState) -> [u8; 64] {
+        hasher.update(&format!("{:?}", self).as_bytes());
+        hasher.finalize().as_bytes().try_into().unwrap()
+    }
 }
diff --git a/src/poly/domain.rs b/src/poly/domain.rs
index e8cc41ee..c9a9dff8 100644
--- a/src/poly/domain.rs
+++ b/src/poly/domain.rs
@@ -8,6 +8,9 @@ use super::{Coeff, ExtendedLagrangeCoeff, LagrangeCoeff, Polynomial, Rotation};
 use ff::{Field, PrimeField};
 use std::marker::PhantomData;
 
+use blake2b_simd::State as Blake2bState;
+use std::convert::TryInto;
+
 /// This structure contains precomputed constants and other details needed for
 /// performing operations on an evaluation domain of size $2^k$ and an extended
 /// domain of size $2^{k} * j$ with $j \neq 0$.
@@ -376,4 +379,18 @@ impl<G: Group> EvaluationDomain<G> {
     pub fn get_quotient_poly_degree(&self) -> usize {
         self.quotient_poly_degree as usize
     }
+
+    /// Hashes the constants in the domain which influence the proof into a u64
+    pub fn hash(&self, hasher: &mut Blake2bState) -> [u8; 64] {
+        hasher.update(b"k");
+        hasher.update(&self.k.to_le_bytes());
+
+        hasher.update(b"extended_k");
+        hasher.update(&self.extended_k.to_le_bytes());
+
+        hasher.update(b"omega");
+        hasher.update(&self.omega.to_bytes());
+
+        hasher.finalize().as_bytes().try_into().unwrap()
+    }
 }